Ultrafilters on metric spaces

Abstract

Let X be an unbounded metric space, B(x,r)={y∈X:d(x,y)≤r} for all x∈X and r≥0. We endow X with the discrete topology and identify the Stone–Čech compactification βX of X with the set of all ultrafilters on X. Our aim is to reveal some features of algebra in βX similar to the algebra in the Stone–Čech compactification of a discrete semigroup [6].We denote X#={p∈βX:each P∈p is unbounded in X} and, for p,q∈X#, write p∥q if and only if there is r≥0 such that B(Q,r)∈p for each Q∈q, where B(Q,r)=⋃x∈QB(x,r). A subset S⊆X# is called invariant if p∈S and q∥p imply q∈S. We characterize the minimal closed invariant subsets of X, the closure of the set K(X#)=⋃{M:M is a minimal closed invariant subset of X#}, and find the number of all minimal closed invariant subsets of X#.For a subset Y⊆X and p∈X#, we denote △p(Y)=Y#∩{q∈X#:p∥q} and say that a subset S⊆X# is an ultracompanion of Y if S=△p(Y) for some p∈X#. We characterize large, thick, prethick, small, thin and asymptotically scattered spaces in terms of their ultracompanions.